Singularity formation for the higher dimensional Skyrme model in the strong field limit

TL;DR

This paper demonstrates the formation of singularities in a higher-dimensional Skyrme model by constructing a stable self-similar solution in the strong field limit, overcoming technical challenges posed by derivative nonlinearities.

Contribution

It provides the first explicit construction and stability analysis of singularity formation in a $(5+1)$-dimensional Skyrme model, extending understanding beyond the well-studied $(3+1)$-dimensional case.

Findings

Existence of an explicit self-similar solution in the strong field limit.

Asymptotic stability of this solution within backwards light cones.

Overcoming technical difficulties caused by derivative nonlinearities using structural properties.

Abstract

This paper concerns the formation of singularities in the classical $(5+1)$-dimensional, co-rotational Skyrme model. While it is well established that blowup is excluded in $(3+1)$-dimensions, nothing appears to be known in the higher dimensional case. We prove that the model, in the so-called strong field limit, admits an explicit self-similar solution which is asymptotically stable within backwards light cones. From a technical point of view, the main obstacle to this result is the presence of derivative nonlinearities in the corresponding evolution equation. These introduce first order terms in the linearized flow which render standard techniques useless. We demonstrate how this problem can be bypassed by using structural properties of the Skyrme model.